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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p><dfn class="terminology">Step 1: Separating variables.</dfn> To find <span class="process-math">\(u(x,t)\text{,}\)</span> we start by making a basic assumption about the form of the solutions: <span class="process-math">\(u(x,t)\)</span> is a product of two functions, one depending only on <span class="process-math">\(x\)</span> and the other depending only on <span class="process-math">\(t\text{;}\)</span> thus</p>
<div class="displaymath process-math" data-contains-math-knowls="        ">
\begin{equation*}
u(x,t)=X(x)T(t).\label{separation}
\end{equation*}
</div>
<p class="continuation">Substituting from Eq. <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "separation" missing or not unique]</code> for <span class="process-math">\(u\)</span> in the differential equation <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "heatpde" missing or not unique]</code> yields</p>
<div class="displaymath process-math" data-contains-math-knowls="        ">
\begin{equation*}
XT'=\alpha^2X''T,\quad\to\quad \frac{X''}{X}=\frac{1}{\alpha^2}\frac{T'}{T},\label{sepa}
\end{equation*}
</div>
<p class="continuation">in which the variables are separated; that is, the left side depends only on <span class="process-math">\(x\)</span> and the right side only on <span class="process-math">\(t\text{.}\)</span> The only way to make it happen is the radio equals a constant. If we call this separation constant <span class="process-math">\(-\lambda\text{,}\)</span> then</p>
<div class="displaymath process-math" data-contains-math-knowls="        ">
\begin{equation*}
\frac{X''}{X}=\frac{1}{\alpha^2}\frac{T'}{T}=-\lambda,
\end{equation*}
</div>
<p class="continuation">We end up with 2 ODEs,</p>
<div class="displaymath process-math" data-contains-math-knowls="        ">
\begin{equation*}
\begin{aligned}
X''+\lambda X &amp;=&amp; 0\label{ode1}\\
T' + \alpha^2\lambda T &amp;=&amp; 0 \label{ode2}
\end{aligned}
\end{equation*}
</div>
<p class="continuation">The assumption <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "separation" missing or not unique]</code> has led to the replacement of the partial differential equation <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "heatpde" missing or not unique]</code> by two ODEs <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "ode1" missing or not unique]</code> and <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "ode2" missing or not unique]</code>. Each of these equations can be readily solved for any value of <span class="process-math">\(\lambda\text{.}\)</span> The product of two solutions of Eq. <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "ode1" missing or not unique]</code> and <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "ode2" missing or not unique]</code>, respectively, provides a solution of the partial differential equation <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "heatpde" missing or not unique]</code>.</p>
<span class="incontext"><a href="sec7_7.html#p-403" class="internal">in-context</a></span>
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